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Determining the action dimension of an Artin group by using its complex of abelian subgroups. (English) Zbl 1433.20011
Summary: Suppose that $$(W,S)$$ is a Coxeter system with associated Artin group $$A$$ and with a simplicial complex $$L$$ as its nerve. We define the notion of a ‘standard abelian subgroup’ in $$A$$. The poset of such subgroups in $$A$$ is parameterized by the poset of simplices in a certain subdivision $$L_\oslash$$ of $$L$$. This complex of standard abelian subgroups is used to generalize an earlier result from the case of right-angled Artin groups to case of general Artin groups, by calculating, in many instances, the smallest dimension of a manifold model for $$BA$$. (This is the ’action dimension’ of $$A$$ denoted $$\operatorname{actdim}A$$.) If $$H_d(L;\mathbb{Z}/2)\neq 0$$, where $$d=\dim L$$, then $$\operatorname{actdim}A\geqslant 2d+2$$. Moreover, when the $$K(\pi,1)$$-Conjecture holds for $$A$$, the inequality is an equality.
MSC:
 20F36 Braid groups; Artin groups 20F55 Reflection and Coxeter groups (group-theoretic aspects) 20F65 Geometric group theory 57S30 Discontinuous groups of transformations 57Q35 Embeddings and immersions in PL-topology 20J06 Cohomology of groups 32S22 Relations with arrangements of hyperplanes
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